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Properties

Label 167310.fg
Number of curves $6$
Conductor $167310$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fg1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 167310.fg have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(11\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 167310.fg do not have complex multiplication.

Modular form 167310.2.a.fg

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} + q^{11} + q^{16} - 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 167310.fg

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
167310.fg1 167310l5 \([1, -1, 1, -260220317, 1615762010841]\) \(553808571467029327441/12529687500\) \(44088759717904687500\) \([2]\) \(23592960\) \(3.2935\)  
167310.fg2 167310l4 \([1, -1, 1, -17985857, -29290444959]\) \(182864522286982801/463015182960\) \(1629231786288773512560\) \([2]\) \(11796480\) \(2.9470\)  
167310.fg3 167310l3 \([1, -1, 1, -16282337, 25188806049]\) \(135670761487282321/643043610000\) \(2262705690738417210000\) \([2, 2]\) \(11796480\) \(2.9470\)  
167310.fg4 167310l6 \([1, -1, 1, -7916837, 51031508649]\) \(-15595206456730321/310672490129100\) \(-1093176886356104709545100\) \([2]\) \(23592960\) \(3.2935\)  
167310.fg5 167310l2 \([1, -1, 1, -1559057, -70453119]\) \(119102750067601/68309049600\) \(240362042099839545600\) \([2, 2]\) \(5898240\) \(2.6004\)  
167310.fg6 167310l1 \([1, -1, 1, 387823, -8931711]\) \(1833318007919/1070530560\) \(-3766922728959836160\) \([2]\) \(2949120\) \(2.2538\) \(\Gamma_0(N)\)-optimal